Right, I hope none of this sounds like I'm trying to teach you to "suck eggs", and if I'm wrong on anything someone please correct me, as it's all coming straight out of my head
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First of all, you need to decide which make of CPU to go with. AMD used to work in conjunction with NVIDIA graphics cards, but I think they now own Radeon - so their CPU's work better with Radeon GFX cards (so I'm told). If you go for an Intel processor, then I don't think it matters what make GFX card you get.
Next thing you need to decide is how many cores you want your CPU to have. When I bought my system, I went for the quad core option because I use a lot of old software, so they only use one core, which the more cores you go, the slower the processor they are using gets. However, if you are mostly using new programs that support multi-core processors, the the more cores the better.
For example, on a 2.7 GHz quad-core CPU, old programs are only using a 0.675GHz processor ... if I'd gone for an eight-core processor, they would only use a speed of 0.3375GHz - marginally faster than my first PC I bought back in 2001!!!
Next is RAM, I would say start with a minimum of 8GB and make sure the system is capable of further expansion. 8GB should do you fine for quite a while, but leaves you room to expand at a later date in case you need it.
For the GFX card, don't just go for the one with the most RAM. The more expensive ones with lesser RAM than a "similar looking" one will probably perform a lot better (and last longer).
I'd definitely consider systems with Solid State Hard Drives (SSD) using SATA3 and which also have USB3 ports. My system has a 120GB SSD for the OS and speed sensitive / processor intensive games / programs. I also have a 2TB 720RPM SATA3 HDD which is split into two partitions - F for files (i.e. your documents folder), and P for Programs (when installing a program / game that you don't want on the SSD, just change the "C" in the path to "P" and leave the rest of the path as it is).
Oh - if your old system used drives (HDD / CD / DVD / etc.) that connect via IDE ports, and you want to use them / connect the HDD to recover your files, make sure the motherboard has IDE ports! - I didn't check this when I bought mine and found it only had SATA 3 ports! I ended up buying a cheap HDD USB caddy and using that to get my old files.
Here's a couple of examples from where I got my desktop PC from...
Quad-core 3.4GHz Intel CPU, 8GB Ram, Barebones kit - £459.99 inc VAT - This system is not a bad starting point - just remember that you need to add a GFX card, Hard Drive, and DVD drive. This one has the same case as mine, which has two fans on the side which you can switch off if needed
Eight-core 4.2GHz AMD CPU, 8GB RAM, Barebones kit - £389.99 inc VAT - This system is even cheaper, but with better CPU speed. As above, you'll need to add GFX card, HDD, DVD drive, etc - but at least you can afford to pick a decent GFX card, some decent drives, and still have cash left over for a new keyboard, 1080p monitor and some decent speakers (and a decent headset too!)